Mapping class group of a surface

Let ${\displaystyle S}$ be a connected, closed, orientable surface and ${\displaystyle \mathrm {Homeo} ^{+}(S)}$ the group of orientation-preserving, or positive, homeomorphisms of ${\displaystyle S}$. This group has a natural topology, the compact-open topology. It can be defined easily by a distance function: if we are given a metric ${\displaystyle d}$ on ${\displaystyle S}$ inducing its topology then the function defined by

${\displaystyle \delta (f,g)=\sup _{x\in S}\left(d(f(x),g(x)\right)}$

is a distance inducing the compact-open topology on ${\displaystyle \mathrm {Homeo} ^{+}(S)}$. The connected component of the identity for this topology is denoted ${\displaystyle \mathrm {Homeo} _{0}(S)}$. By definition it is equal to the homeomorphisms of ${\displaystyle S}$ which are isotopic to the identity. It is a normal subgroup of the group of positive homeomorphisms, and the mapping class group of ${\displaystyle S}$ is the group

${\displaystyle \mathrm {Mod} (S)=\mathrm {Homeo} ^{+}(S)/\mathrm {Homeo} _{0}(S)}$.

This is a countable group.

If we modify the definition to include all homeomorphisms we obtain the extended mapping class group ${\displaystyle \mathrm {Mod} ^{\pm }(S)}$, which contains the mapping class group as a subgroup of index 2.

This definition can also be made in the differentiable category: if we replace all instances of "homeomorphism" above with "diffeomorphism" we obtain the same group, that is the inclusion ${\displaystyle \mathrm {Diff} ^{+}(S)\subset \mathrm {Homeo} ^{+}(S)}$ induces an isomorphism between the quotients by their respective identity components.

Suppose that ${\displaystyle S}$ is the unit sphere in ${\displaystyle \mathbb {R} ^{3}}$. Then any homeomorphism of ${\displaystyle S}$ is isotopic to either the identity or to the restriction to ${\displaystyle S}$ of the symmetry in the plane ${\displaystyle z=0}$. The latter is not orientation-preserving and we see that the mapping class group of the sphere is trivial, and its extended mapping class group is ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$, the cyclic group of order 2.

The mapping class group of the torus ${\displaystyle \mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ is naturally identified with the modular group ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}$. It is easy to construct a morphism ${\displaystyle \Phi$ :\mathrm {SL} _{2}(\mathbb {Z} )\to \mathrm {Mod} (\mathbb {T} ^{2})} : every ${\displaystyle A\in \mathrm {SL} _{2}(\mathbb {Z} )}$ induces a diffeomorphism of ${\displaystyle \mathbb {T} ^{2}}$ via ${\displaystyle x+\mathbb {Z} ^{2}\mapsto Ax+\mathbb {Z} ^{2}}$. The action of diffeomorphisms on the first homology group of ${\displaystyle \mathbb {T} ^{2}}$ gives a left-inverse ${\displaystyle \Pi }$ to the morphism ${\displaystyle \Phi }$ (proving in particular that it is injective) and it can be checked that ${\displaystyle \Pi }$ is injective, so that ${\displaystyle \Pi ,\Phi }$ are inverse isomorphisms between ${\displaystyle \mathrm {Mod} (\mathbb {T} ^{2})}$ and ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}$.[3] In the same way, the extended mapping class group of ${\displaystyle \mathbb {T} ^{2}}$ is ${\displaystyle \mathrm {GL} _{2}(\mathbb {Z} )}$.

In the case where ${\displaystyle S}$ is a compact surface with a non-empty boundary ${\displaystyle \partial S}$ then the definition of the mapping class group needs to be more precise. The group ${\displaystyle \mathrm {Homeo} ^{+}(S,\partial S)}$ of homeomorphisms relative to the boundary is the subgroup of ${\displaystyle \mathrm {Homeo} ^{+}(S)}$ which restrict to the identity on the boundary, and the subgroup ${\displaystyle \mathrm {Homeo} _{0}(S,\partial S)}$ is the connected component of the identity. The mapping class group is then defined as

${\displaystyle \mathrm {Mod} (S)=\mathrm {Homeo} ^{+}(S,\partial S)/\mathrm {Homeo} _{0}(S,\partial S)}$.

A surface with punctures is a compact surface with a finite number of points removed ("punctures"). The mapping class group of such a surface is defined as above (note that the mapping classes are allowed to permute the punctures, but not the boundary components).

Any annulus is homeomorphic to the subset ${\displaystyle A_{0}=\{1\leq |z|\leq 2\}}$ of ${\displaystyle \mathbb {C} }$. One can define a diffeomorphism ${\displaystyle \tau _{0}}$ by the following formula:

${\displaystyle \tau _{0}(z)=e^{2i\pi |z|}z}$

which is the identity on both boundary components ${\displaystyle \{|z|=1\},\{|z|=2\}}$. The mapping class group of ${\displaystyle A}$ is then generated by the class of ${\displaystyle \tau _{0}}$.

Braid groups can be defined as the mapping class groups of a disc with punctures. More precisely, the braid group on n strands is naturally isomorphic to the mapping class group of a disc with n punctures.[4]

If ${\displaystyle S}$ is closed and ${\displaystyle f}$ is a homeomorphism of ${\displaystyle S}$ then we can define an automorphism ${\displaystyle f_{*}}$ of the fundamental group ${\displaystyle \pi _{1}(S,x_{0})}$ as follows: fix a path ${\displaystyle \gamma }$ between ${\displaystyle x_{0}}$ and ${\displaystyle f(x_{0})}$ and for a loop ${\displaystyle \alpha }$ based at ${\displaystyle x_{0}}$ representing an element ${\displaystyle [\alpha ]\in \pi _{1}(S,x_{0})}$ define ${\displaystyle f_{*}([\alpha ])}$ to be the element of the fundamental group associated to the loop ${\displaystyle {\bar {\gamma }}*f(\alpha )*\gamma }$. This automorphism depends on the choice of ${\displaystyle \gamma }$, but only up to conjugation. Thus we get a well-defined map from ${\displaystyle \mathrm {Homeo} (S)}$ to the outer automorphism group ${\displaystyle \mathrm {Out} (\pi _{1}(S,x_{0}))}$. This map is a morphism and its kernel is exactly the subgroup ${\displaystyle \mathrm {Homeo} _{0}(S)}$. The Dehn–Nielsen–Baer theorem states that it is in addition surjective.[5] In particular, it implies that:

The extended mapping class group ${\displaystyle \mathrm {Mod} ^{\pm }(S)}$ is isomorphic to the outer automorphism group ${\displaystyle \mathrm {Out} (\pi _{1}(S))}$.

The image of the mapping class group is an index 2 subgroup of the outer automorphism group, which can be characterised by its action on homology.

The conclusion of the theorem does not hold when ${\displaystyle S}$ has a non-empty boundary (except in a finite number of cases). In this case the fundamental group is a free group and the outer automorphism group Out(Fn) is strictly larger than the image of the mapping class group via the morphism defined in the previous paragraph. The image is exactly those outer automorphisms which preserve each conjugacy class in the fundamental group corresponding to a boundary component.

This is an exact sequence relating the mapping class group of surfaces with the same genus and boundary but a different number of punctures. It is a fundamental tool which allows to use recursive arguments in the study of mapping class groups. It was proven by Joan Birman in 1969.[6] The exact statement is as follows.[7]

Let ${\displaystyle S}$ be a compact surface and ${\displaystyle x\in S}$. There is an exact sequence

${\displaystyle 1\to \pi _{1}(S,x)\to \mathrm {Mod} (S\setminus \{x\})\to \mathrm {Mod} (S)\to 1}$.

In the case where ${\displaystyle S}$ itself has punctures the mapping class group ${\displaystyle \mathrm {Mod} (S\setminus \{x\})}$ must be replaced by the finite-index subgroup of mapping classes fixing ${\displaystyle x}$.

If ${\displaystyle c}$ is an oriented simple closed curve on ${\displaystyle S}$ and one chooses a closed tubular neighbourhood ${\displaystyle A}$ then there is a homeomorphism ${\displaystyle f}$ from ${\displaystyle A}$ to the canonical annulus ${\displaystyle A_{0}}$ defined above, sending ${\displaystyle c}$ to a circle with the counterclockwise orientation. This is used to define a homeomorphism ${\displaystyle \tau _{c}}$ of ${\displaystyle S}$ as follows: on ${\displaystyle S\setminus A}$ it is the identity, and on ${\displaystyle A}$ it is equal to ${\displaystyle f^{-1}\circ \tau _{0}\circ f}$. The class of ${\displaystyle \tau _{c}}$ in the mapping class group ${\displaystyle \mathrm {Mod} (S)}$ does not depend on the choice of ${\displaystyle f}$ made above, and the resulting element is called the Dehn twist about ${\displaystyle c}$. If ${\displaystyle c}$ is not null-homotopic this mapping class is nontrivial, and more generally the Dehn twists defined by two non-homotopic curves are distinct elements in the mapping class group.

In the mapping class group of the torus identified with ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}$ the Dehn twists correspond to unipotent matrices. For example, the matrix

${\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}$

corresponds to the Dehn twist about a horizontal curve in the torus.

There is a classification of the mapping classes on a surface, originally due to Nielsen and rediscovered by Thurston, which can be stated as follows. An element ${\displaystyle g\in \mathrm {Mod} (S)}$ is either:

• of finite order (i.e. there exists ${\displaystyle n>0}$ such that ${\displaystyle g^{n}}$ is the identity),
• reducible: there exists a set of disjoint closed curves on ${\displaystyle S}$ which is preserved by the action of ${\displaystyle g}$;
• or pseudo-Anosov.

The main content of the theorem is that a mapping class which is neither of finite order nor reducible must be pseudo-Anosov, which can be defined explicitly by dynamical properties.[8]

The study of pseudo-Anosov diffeomorphisms of a surface is fundamental. They are the most interesting diffeomorphisms, since finite-order mapping classes are isotopic to isometries and thus well understood, and the study of reducible classes indeed essentially reduces to the study of mapping classes on smaller surfaces which may themselves be either finite order or pseudo-Anosov.

Pseudo-Anosov mapping classes are "generic" in the mapping class group in various ways. For example, a random walk on the mapping class group will end on a pseudo-Anosov element with a probability tending to 1 as the number of steps grows.

Given a punctured surface ${\displaystyle S}$ (usually without boundary) the Teichmüller space ${\displaystyle T(S)}$ is the space of marked complex (equivalently, conformal or complete hyperbolic) structures on ${\displaystyle S}$. These are represented by pairs ${\displaystyle (X,f)}$ where ${\displaystyle X}$ is a Riemann surface and ${\displaystyle f:S\to X}$ a homeomorphism, modulo a suitable equivalence relation. There is an obvious action of the group ${\displaystyle \mathrm {Homeo} ^{+}(S)}$ on such pairs, which descends to an action of ${\displaystyle \mathrm {Mod} (S)}$ on Teichmüller space.

This action has many interesting properties; for example it is properly discontinuous (though not free). It is compatible with various geometric structures (metric or complex) with which ${\displaystyle T(S)}$ can be endowed. In particular, the Teichmüller metric can be used to establish some large-scale properties of the mapping class group, for example that the maximal quasi-isometrically embedded flats in ${\displaystyle \mathrm {Mod} (S)}$ are of dimension ${\displaystyle 3g-3+k}$.[9]

The action extends to the Thurston boundary of Teichmüller space, and the Nielsen-Thurston classification of mapping classes can be seen in the dynamical properties of the action on Teichmüller space together with its Thurston boundary. Namely:[10]

• Finite-order elements fix a point inside Teichmüller space (more concretely this means that any mapping class of finite order in ${\displaystyle \mathrm {Mod} (S)}$ can be realised as an isometry for some hyperbolic metric on ${\displaystyle S}$);
• Pseudo-Anosov classes fix the two points on the boundary corresponding to their stable and unstable foliation and the action is minimal (has a dense orbit) on the boundary;
• Reducible classes do not act minimally on the boundary.

The curve complex of a surface ${\displaystyle S}$ is a complex whose vertices are isotopy classes of simple closed curves on ${\displaystyle S}$. The action of the mapping class groups ${\displaystyle \mathrm {Mod} (S)}$ on the vertices carries over to the full complex. The action is not properly discontinuous (the stabiliser of a simple closed curve is an infinite group).

This action, together with combinatorial and geometric properties of the curve complex, can be used to prove various properties of the mapping class group.[11] In particular, it explains some of the hyperbolic properties of the mapping class group: while as mentioned in the previous section the mapping class group is not an hyperbolic group it has some properties reminiscent of those.

The pants complex of a compact surface ${\displaystyle S}$ is a complex whose vertices are the pants decompositions of ${\displaystyle S}$ (isotopy classes of maximal systems of disjoint simple closed curves). The action of ${\displaystyle \mathrm {Mod} (S)}$ extends to an action on this complex. This complex is quasi-isometric to Teichmüller space endowed with the Weil–Petersson metric.[12]

The stabilisers of the mapping class group's action on the curve and pants complexes are quite large. The markings complex is a complex whose vertices are markings of ${\displaystyle S}$, which are acted upon by, and have trivial stabilisers in, the mapping class group ${\displaystyle \mathrm {Mod} (S)}$. It is (in opposition to the curve or pants complex) a locally finite complex which is quasi-isometric to the mapping class group.[13]

A marking[lower-alpha 1] is determined by a pants decomposition ${\displaystyle \alpha _{1},\ldots ,\alpha _{\xi }}$ and a collection of transverse curves ${\displaystyle \beta _{1},\ldots ,\beta _{\xi }}$ such that every one of the ${\displaystyle \beta _{i}}$ intersects at most one of the ${\displaystyle \alpha _{i}}$, and this "minimally" (this is a technical condition which can be stated as follows: if ${\displaystyle \alpha _{i},\beta _{i}}$ are contained in a subsurface homeomorphic to a torus then they intersect once, and if the surface is a four-holed sphere they intersect twice). Two distinct markings are joined by an edge if they differ by an "elementary move", and the full complex is obtained by adding all possible higher-dimensional simplices.

The mapping class group is generated by the subset of Dehn twists about all simple closed curves on the surface. The Dehn–Lickorish theorem states that it is sufficient to select a finite number of those to generate the mapping class group.[14] This generalises the fact that ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}$ is generated by the matrices

${\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\1&1\end{pmatrix}}}$.

In particular, the mapping class group of a surface is a finitely generated group.

The smallest number of Dehn twists that can generate the mapping class group of a closed surface of genus ${\displaystyle g\geq 2}$ is ${\displaystyle 2g+1}$; this was proven later by Humphries.

It is possible to prove that all relations between the Dehn twists in a generating set for the mapping class group can be written as combinations of a finite number among them. This means that the mapping class group of a surface is a finitely presented group.

One way to prove this theorem is to deduce it from the properties of the action of the mapping class group on the pants complex: the stabiliser of a vertex is seen to be finitely presented, and the action is cofinite. Since the complex is connected and simply connected it follows that the mapping class group must be finitely generated. There are other ways of getting finite presentations, but in practice the only one to yield explicit relations for all geni is that described in this paragraph with a slightly different complex instead of the curve complex, called the cut system complex.[15]

An example of a relation between Dehn twists occurring in this presentation is the lantern relation.

There are other interesting systems of generators for the mapping class group besides Dehn twists. For example, ${\displaystyle \mathrm {Mod} (S)}$ can be generated by two elements[16] or by involutions.[17]

If ${\displaystyle S}$ is a surface of genus ${\displaystyle g}$ with ${\displaystyle b}$ boundary components and ${\displaystyle k}$ punctures then the virtual cohomological dimension of ${\displaystyle \mathrm {Mod} (S)}$ is equal to ${\displaystyle 4g-4+b+k}$.

The first homology of the mapping class group is finite[18] and it follows that the first cohomology group is finite as well.

As singular homology is functorial, the mapping class group ${\displaystyle \mathrm {Mod} (S)}$ acts by automorphisms on the first homology group ${\displaystyle H_{1}(S)}$. This is a free abelian group of rank ${\displaystyle 2g}$ if ${\displaystyle S}$ is closed of genus ${\displaystyle g}$. This action thus gives a linear representation ${\displaystyle \mathrm {Mod} (S)\to \mathrm {GL} _{2g}(\mathbb {Z} )}$.

This map is in fact a surjection with image equal to the integer points ${\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )}$ of the symplectic group. This comes from the fact that the intersection number of closed curves induces a symplectic form on the first homology, which is preserved by the action of the mapping class group. The surjectivity is proven by showing that the images of Dehn twists generate ${\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )}$.[19]

The kernel of the morphism ${\displaystyle \mathrm {Mod} (S)\to \mathrm {Sp} _{2g}(\mathbb {Z} )}$ is called the Torelli group of ${\displaystyle S}$. It is a finitely generated, torsion-free subgroup[20] and its study is of fundamental importance for its bearing on both the structure of the mapping class group itself (since the arithmetic group ${\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )}$ is comparatively very well understood, a lot of facts about ${\displaystyle \mathrm {Mod} (S)}$ boil down to a statement about its Torelli subgroup) and applications to 3-dimensional topology and algebraic geometry.

An example of application of the Torelli subgroup is the following result:

The mapping class group is residually finite.

The proof proceeds first by using residual finiteness of the linear group ${\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )}$, and then, for any nontrivial element of the Torelli group, constructing by geometric means subgroups of finite index which does not contain it.[21]

An interesting class of finite-index subgroups is given by the kernels of the morphisms:

${\displaystyle \Phi _{n}:\mathrm {Mod} (S)\to \mathrm {Sp} _{2g}(\mathbb {Z} )\to \mathrm {Sp} _{2g}(\mathbb {Z} /n\mathbb {Z} )}$

The kernel of ${\displaystyle \Phi _{n}}$ is usually called a congruence subgroup of ${\displaystyle \mathrm {Mod} (S)}$. It is a torsion-free group for all ${\displaystyle n\geq 3}$ (this follows easily from a classical result of Minkowski on linear groups and the fact that the Torelli group is torsion-free).

The mapping class group has only finitely many classes of finite groups, as follows from the fact that the finite-index subgroup ${\displaystyle \ker(\Phi _{3})}$ is torsion-free, as discussed in the previous paragraph. Moreover, this also implies that any finite subgroup of ${\displaystyle \mathrm {Mod} (S)}$ is a subgroup of the finite group ${\displaystyle \mathrm {Mod} (S)/\ker(\Phi _{3})\cong \mathrm {Sp} _{2g}(\mathbb {Z} /3)}$.

A bound on the order of finite subgroups can also be obtained through geometric means. The solution to the Nielsen realisation problem implies that any such group is realised as the group of isometries of an hyperbolic surface of genus ${\displaystyle g}$. Hurwitz's bound then implies that the maximal order is equal to ${\displaystyle 84(g-1)}$.

The mapping class groups satisfy the Tits alternative: that is, any subgroup of it either contains a non-abelian free subgroup or it is virtually solvable (in fact abelian).[22]

Any subgroup which is not reducible (that is it does not preserve a set of isotopy class of disjoint simple closed curves) must contain a pseudo-Anosov element.[23]